Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.
Triangles, parallelograms, and trapezoids are all two-dimensional shapes that show up in the world around us. In this lesson, we'll look at the areas of these shapes and how they relate to each other.
Three Different Shapes
Let's talk about shapes, three in particular! They are the triangle, the parallelogram, and the trapezoid. These three shapes are related in many ways, including their area formulas. The area of a two-dimensional shape is the amount of space inside that shape. Before we get to those relationships, let's take a moment to define each of these shapes and their area formulas.
Let's first look at parallelograms. A parallelogram is a four-sided, two-dimensional shape in which opposite sides are parallel and have equal length.
To find the area of a parallelogram, we simply multiply the base times the height.
Now, let's look at triangles. You've probably heard of a triangle. A triangle is a two-dimensional shape with three sides and three angles.
To find the area of a triangle, we take one half of its base multiplied by its height.
Finally, let's look at trapezoids. A trapezoid is lesser known than a triangle, but still a common shape. A trapezoid is a four-sided, two-dimensional shape with two parallel sides. Trapezoids have two bases. Those are the sides that are parallel.
To find the area of a trapezoid, we multiply one half times the sum of the bases times the height.
Alright! Now that we got all the definitions and formulas out of the way, let's look at how these three shapes' areas are related. To get started, let me ask you: do you like puzzles? That probably sounds odd, but as it turns out, we can create parallelograms using triangles or trapezoids as puzzle pieces. This fact will help us to illustrate the relationship between these shapes' areas.
Let's first look at the relationship between parallelograms and triangles. First, let's consider triangles and parallelograms. Notice that if we cut a parallelogram diagonally to divide it in half, we form two triangles, with the same base and height as the parallelogram.
We see that each triangle takes up precisely one half of the parallelogram. From this, we see that the area of a triangle is one half the area of a parallelogram, or the area of a parallelogram is two times the area of a triangle. By looking at a parallelogram as a puzzle put together by two equal triangle pieces, we have the relationship between the areas of these two shapes, like you can see in all these equations.
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Now, let's look at the relationship between parallelograms and trapezoids. In the same way that we can create a parallelogram from two triangles, we can also create a parallelogram from two trapezoids. To do this, we flip a trapezoid upside down and line it up next to itself as shown.
From the image, we see that we can create a parallelogram from two trapezoids, or we can divide any parallelogram into two equal trapezoids. When we do this, the base of the parallelogram has length b 1 + b 2, and the height is the same as the trapezoids, so the area of the parallelogram is (b 1 + b 2)*h.
Since the two trapezoids of the same size created this parallelogram, the area of one of those trapezoids is one half the area of the parallelogram. This is how we get the area of a trapezoid: 1/2(b 1 + b 2)*h. We see yet another relationship between these shapes.
Let's take a few moments to review what we've learned about the relationships between the area formulas of triangles, parallelograms, and trapezoids. A parallelogram is a four-sided, two-dimensional shape with opposite sides that are parallel and have equal length. A triangle is a two-dimensional shape with three sides and three angles. A trapezoid is a two-dimensional shape with two parallel sides. The area formulas of these three shapes are shown right here:
We see that we can create a parallelogram from two triangles or from two trapezoids, like a puzzle. In doing this, we illustrate the relationship between the area formulas of these three shapes. These relationships make us more familiar with these shapes and where their area formulas come from.
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